Question: $ C = \left[\begin{array}{rr}-2 & -2 \\ -1 & 1\end{array}\right]$ $ E = \left[\begin{array}{rrr}2 & 1 & -2 \\ 3 & 5 & 4\end{array}\right]$ What is $ C E$ ?
Because $ C$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C E = \left[\begin{array}{rr}{-2} & {-2} \\ {-1} & {1}\end{array}\right] \left[\begin{array}{rrr}{2} & \color{#DF0030}{1} & \color{#9D38BD}{-2} \\ {3} & \color{#DF0030}{5} & \color{#9D38BD}{4}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{2}+{-2}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{2}+{-2}\cdot{3} & ? & ? \\ {-1}\cdot{2}+{1}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{2}+{-2}\cdot{3} & {-2}\cdot\color{#DF0030}{1}+{-2}\cdot\color{#DF0030}{5} & ? \\ {-1}\cdot{2}+{1}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-2}\cdot{2}+{-2}\cdot{3} & {-2}\cdot\color{#DF0030}{1}+{-2}\cdot\color{#DF0030}{5} & {-2}\cdot\color{#9D38BD}{-2}+{-2}\cdot\color{#9D38BD}{4} \\ {-1}\cdot{2}+{1}\cdot{3} & {-1}\cdot\color{#DF0030}{1}+{1}\cdot\color{#DF0030}{5} & {-1}\cdot\color{#9D38BD}{-2}+{1}\cdot\color{#9D38BD}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-10 & -12 & -4 \\ 1 & 4 & 6\end{array}\right] $